\chapter{Mathematical Models and Algorithms}
\label{chap:mathematical_models}

\section{Introduction}

Building upon the formal framework established in Chapter \ref{chap:formal_model}, this chapter develops detailed mathematical models for the key algorithmic components of \ClaudeCode{}. We present rigorous mathematical treatments of tool selection, context management, concurrency control, and code generation, accompanied by algorithmic analysis and performance guarantees.

The mathematical models developed here serve dual purposes: providing theoretical understanding of system behavior and guiding practical implementation decisions. Each model is accompanied by algorithmic instantiation and complexity analysis.

\section{Contextual Bandit Framework for Tool Selection}

Tool selection in \ClaudeCode{} exhibits the characteristics of a contextual bandit problem, where the system must balance exploration of new tools with exploitation of known effective tools based on current context.

\subsection{Problem Formulation}

\begin{definition}[Contextual Tool Selection Bandit]
\label{def:contextual_bandit}
At each time step $t$, the system:
\begin{enumerate}
    \item Observes context vector $x_t \in \mathcal{X} \subseteq \R^d$
    \item Selects tool-parameter pair $(t_i, \theta_i) \in \Tools \times \Theta$
    \item Receives reward $r_t \in [0, 1]$ drawn from distribution with mean $\mu_{t_i,\theta_i}(x_t)$
    \item Updates its model of tool effectiveness
\end{enumerate}
\end{definition}

\begin{definition}[Context Feature Extraction]
\label{def:context_features}
The context vector $x_t$ is constructed from:
\begin{align}
x_t = \phi(&\text{task\_type}, \text{codebase\_features}, \text{recent\_history}, \\
&\text{resource\_constraints}, \text{user\_preferences})
\end{align}
where $\phi: \mathcal{C} \rightarrow \R^d$ is a feature extraction function.
\end{definition}

\subsection{Linear Contextual Bandit Model}

We model tool effectiveness using a linear contextual bandit framework.

\begin{assumption}[Linear Tool Effectiveness]
\label{ass:linear_effectiveness}
The expected reward for tool $t_i$ with parameters $\theta_i$ in context $x_t$ satisfies:
\begin{equation}
\mu_{t_i,\theta_i}(x_t) = x_t^T \beta_{t_i,\theta_i} + \epsilon_{t_i,\theta_i}(x_t)
\end{equation}
where $\beta_{t_i,\theta_i} \in \R^d$ is the parameter vector and $|\epsilon_{t_i,\theta_i}(x_t)| \leq \epsilon$ for some small $\epsilon > 0$.
\end{assumption}

\begin{definition}[LinUCB Algorithm for Tool Selection]
\label{def:linucb}
The LinUCB algorithm maintains for each tool-parameter pair $(t_i, \theta_i)$:
\begin{align}
A_{t_i,\theta_i}(t) &= \sum_{s=1}^{t-1} x_s x_s^T \mathbf{1}_{a_s = (t_i,\theta_i)} + \lambda I_d \\
b_{t_i,\theta_i}(t) &= \sum_{s=1}^{t-1} x_s r_s \mathbf{1}_{a_s = (t_i,\theta_i)} \\
\hat{\beta}_{t_i,\theta_i}(t) &= A_{t_i,\theta_i}(t)^{-1} b_{t_i,\theta_i}(t) \\
\text{UCB}_{t_i,\theta_i}(t) &= x_t^T \hat{\beta}_{t_i,\theta_i}(t) + \alpha \sqrt{x_t^T A_{t_i,\theta_i}(t)^{-1} x_t}
\end{align}
\end{definition}

\begin{theorem}[LinUCB Regret Bound]
\label{thm:linucb_regret}
Under Assumption \ref{ass:linear_effectiveness}, the LinUCB algorithm achieves cumulative regret bounded by:
\begin{equation}
\text{Regret}(T) = O(d\sqrt{T \log(|\Tools| \cdot |\Theta| \cdot T/\delta)})
\end{equation}
with probability at least $1 - \delta$.
\end{theorem}

\begin{proof}
This follows from the standard LinUCB analysis with multiple arms. See \cite{Li2010} and Appendix \ref{app:proofs} for complete proof.
\end{proof}

\subsection{Thompson Sampling for Tool Selection}

As an alternative to UCB methods, we develop a Thompson sampling approach for tool selection.

\begin{definition}[Bayesian Tool Effectiveness Model]
\label{def:bayesian_tool}
We place a Gaussian prior on tool effectiveness parameters:
\begin{equation}
\beta_{t_i,\theta_i} \sim \mathcal{N}(0, \lambda^{-1} I_d)
\end{equation}
\end{definition}

\begin{algorithm}
\caption{Thompson Sampling for Tool Selection}
\label{alg:thompson_sampling}
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}

\Input{Context $x_t$, tool set $\Tools$, parameter spaces $\{\Theta_i\}$}
\Output{Selected tool-parameter pair $(t^*, \theta^*)$}

\For{each tool-parameter pair $(t_i, \theta_i)$}{
    Compute posterior mean $\hat{\mu}_{t_i,\theta_i} = x_t^T A_{t_i,\theta_i}^{-1} b_{t_i,\theta_i}$\;
    Compute posterior covariance $\Sigma_{t_i,\theta_i} = A_{t_i,\theta_i}^{-1}$\;
    Sample $\tilde{\mu}_{t_i,\theta_i} \sim \mathcal{N}(\hat{\mu}_{t_i,\theta_i}, \sigma^2 x_t^T \Sigma_{t_i,\theta_i} x_t)$\;
}

\Return{$(t^*, \theta^*) = \arg\max_{t_i,\theta_i} \tilde{\mu}_{t_i,\theta_i}$}\;
\end{algorithm}

\begin{theorem}[Thompson Sampling Regret]
\label{thm:thompson_regret}
The Thompson sampling algorithm achieves expected cumulative regret:
\begin{equation}
\E[\text{Regret}(T)] = O(d\sqrt{T})
\end{equation}
which matches the lower bound up to logarithmic factors.
\end{theorem}

\section{Submodular Optimization for Context Selection}

Context selection under token budget constraints exhibits submodular structure, enabling efficient approximation algorithms.

\subsection{Submodular Context Utility}

\begin{definition}[Context Element Value]
\label{def:context_value}
For a context element $v \in V$ and current selection $S$, define:
\begin{equation}
\text{value}(v | S) = \alpha \cdot \text{Relevance}(v | \text{query}) + \beta \cdot \text{Coverage}(v | S) + \gamma \cdot \text{Diversity}(v | S)
\end{equation}
\end{definition}

\begin{definition}[Submodular Context Function]
\label{def:submodular_context}
The context utility function is defined as:
\begin{equation}
f(S) = \sum_{v \in S} \text{Relevance}(v) + \lambda \sqrt{\sum_{v \in S} \text{Coverage}(v)^2} - \mu \sum_{i,j \in S, i \neq j} \text{Overlap}(i,j)
\end{equation}
\end{definition}

\begin{proposition}[Context Function Submodularity]
\label{prop:context_submodular}
Under appropriate parameterization, the function $f(S)$ defined above is monotonic and submodular.
\end{proposition}

\begin{proof}
Monotonicity follows from non-negative relevance scores. Submodularity follows from the diminishing returns property of the coverage and diversity terms.
\end{proof}

\subsection{Greedy Context Selection Algorithm}

\begin{algorithm}
\caption{Greedy Context Selection}
\label{alg:greedy_context}
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}

\Input{Candidate elements $V$, budget $B$, utility function $f$}
\Output{Selected context set $S$}

Initialize $S = \emptyset$, remaining budget $b = B$\;

\While{$b > 0$ and $V \setminus S \neq \emptyset$}{
    $v^* = \arg\max_{v \in V \setminus S, \text{size}(v) \leq b} \frac{f(S \cup \{v\}) - f(S)}{\text{size}(v)}$\;
    
    \If{no such $v^*$ exists}{
        break\;
    }
    
    $S = S \cup \{v^*\}$\;
    $b = b - \text{size}(v^*)$\;
}

\Return{$S$}\;
\end{algorithm}

\begin{theorem}[Greedy Context Selection Approximation]
\label{thm:greedy_context}
Algorithm \ref{alg:greedy_context} achieves a $(1 - 1/e)$-approximation to the optimal context selection under the submodularity assumption.
\end{theorem}

\begin{proof}
This follows from the standard analysis of greedy algorithms for submodular maximization subject to a knapsack constraint.
\end{proof}

\subsection{Dynamic Context Update}

Context selection must adapt as new information becomes available.

\begin{definition}[Dynamic Context Update]
\label{def:dynamic_context}
At each time step, the context selection is updated using:
\begin{equation}
S_{t+1} = \arg\max_{S: |S| \leq B} f_t(S) + \eta \cdot f_{t+1}(S)
\end{equation}
where $f_t$ represents historical value and $f_{t+1}$ represents projected future value.
\end{definition}

\begin{algorithm}
\caption{Online Context Selection}
\label{alg:online_context}
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}

\Input{Current context $S_t$, new elements $V_{\text{new}}$, budget $B$}
\Output{Updated context $S_{t+1}$}

Compute marginal values: $\Delta_v = f(S_t \cup \{v\}) - f(S_t)$ for $v \in V_{\text{new}}$\;

Compute removal benefits: $\Delta_{-u} = f(S_t) - f(S_t \setminus \{u\})$ for $u \in S_t$\;

\While{budget constraints violated or improvement possible}{
    \If{$\max_v \Delta_v > \min_u \Delta_{-u}$ and budget allows}{
        $v^* = \arg\max_v \Delta_v$, $u^* = \arg\min_u \Delta_{-u}$\;
        $S_t = (S_t \setminus \{u^*\}) \cup \{v^*\}$\;
        Update marginal values\;
    }
    \Else{
        break\;
    }
}

\Return{$S_t$}\;
\end{algorithm}

\section{Queueing Models for Concurrency Management}

\ClaudeCode{}'s concurrent tool execution requires careful analysis using queueing theory.

\subsection{System Model}

\begin{definition}[Queueing Network Model]
\label{def:queueing_network}
The system is modeled as a network of queues:
\begin{itemize}
    \item \textbf{Request Queue}: User requests arrive with rate $\lambda$
    \item \textbf{Tool Execution Queues}: $k$ parallel servers with service rates $\mu_1, \ldots, \mu_k$
    \item \textbf{Result Processing Queue}: Single server with rate $\mu_r$
\end{itemize}
\end{definition}

\begin{definition}[Service Time Distribution]
\label{def:service_time}
Tool execution times follow a general distribution with:
\begin{align}
\E[S_i] &= 1/\mu_i \\
\text{Var}[S_i] &= \sigma_i^2 \\
C_i^2 &= \sigma_i^2 \mu_i^2 \quad \text{(coefficient of variation)}
\end{align}
\end{definition}

\subsection{Performance Analysis}

\begin{theorem}[System Stability Condition]
\label{thm:stability_condition}
The system is stable if and only if:
\begin{equation}
\rho = \frac{\lambda}{\min(\sum_{i=1}^k \mu_i, \mu_r)} < 1
\end{equation}
\end{theorem}

\begin{theorem}[Response Time Analysis]
\label{thm:response_time}
For the M/G/k queue model with $k$ parallel tool execution servers, the mean response time is:
\begin{equation}
\E[W] = \E[W_q] + \E[S]
\end{equation}
where the mean waiting time $\E[W_q]$ satisfies the Pollaczek-Khinchine formula:
\begin{equation}
\E[W_q] = \frac{\lambda \E[S^2] P_k}{2(k\mu - \lambda)}
\end{equation}
and $P_k$ is the probability that all servers are busy.
\end{theorem}

\subsection{Optimal Concurrency Level}

\begin{definition}[Cost-Benefit Trade-off]
\label{def:concurrency_tradeoff}
The optimal concurrency level balances throughput benefits against context contention costs:
\begin{equation}
\text{Objective}(k) = \text{Throughput}(k) - C_{\text{context}} \cdot \text{Contention}(k) - C_{\text{resource}} \cdot k
\end{equation}
\end{definition}

\begin{algorithm}
\caption{Adaptive Concurrency Control}
\label{alg:concurrency_control}
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}

\Input{Current concurrency $k$, performance metrics, system constraints}
\Output{Updated concurrency level $k'$}

Measure current performance: $P_{\text{current}} = \text{Objective}(k)$\;

\If{system underutilized and resources available}{
    Test $k + 1$: Estimate $P_{\text{test}} = \text{Objective}(k+1)$\;
    \If{$P_{\text{test}} > P_{\text{current}} + \text{threshold}$}{
        $k' = k + 1$\;
    }
}
\ElseIf{system overloaded or performance degrading}{
    Test $k - 1$: Estimate $P_{\text{test}} = \text{Objective}(k-1)$\;
    \If{$P_{\text{test}} > P_{\text{current}}$}{
        $k' = k - 1$\;
    }
}

\Return{$k'$}\;
\end{algorithm}

\section{Graph-Based Code Understanding}

Code understanding in \ClaudeCode{} leverages graph-theoretic models of codebase structure.

\subsection{Code Dependency Graph}

\begin{definition}[Code Dependency Graph]
\label{def:code_graph}
The code dependency graph $G = (V, E)$ where:
\begin{itemize}
    \item $V$: Set of code entities (functions, classes, modules, files)
    \item $E$: Set of dependency relationships (calls, imports, inheritance)
\end{itemize}
Each vertex $v \in V$ has attributes including type, location, and semantic features.
\end{definition}

\begin{definition}[Graph Metrics]
\label{def:graph_metrics}
Key graph-theoretic measures include:
\begin{align}
\text{PageRank}(v) &= \frac{1-d}{|V|} + d \sum_{u:(u,v) \in E} \frac{\text{PageRank}(u)}{|\text{out}(u)|} \\
\text{Centrality}(v) &= \sum_{s,t \in V} \frac{\sigma_{st}(v)}{\sigma_{st}} \\
\text{Clustering}(v) &= \frac{2|\{(u,w) : u,w \in N(v), (u,w) \in E\}|}{|N(v)|(|N(v)|-1)}
\end{align}
where $d$ is the damping factor, $\sigma_{st}$ is the number of shortest paths from $s$ to $t$, and $N(v)$ is the neighborhood of $v$.
\end{definition}

\subsection{Exploration Strategy}

\begin{definition}[Guided Graph Exploration]
\label{def:graph_exploration}
Code exploration follows a guided random walk with transition probabilities:
\begin{equation}
P(u \rightarrow v) = \frac{\exp(\beta \cdot \text{Utility}(v | \text{context}))}{\sum_{w \in N(u)} \exp(\beta \cdot \text{Utility}(w | \text{context}))}
\end{equation}
where $\beta$ controls exploration vs. exploitation and $\text{Utility}(v | \text{context})$ measures relevance.
\end{definition}

\begin{algorithm}
\caption{Priority-Based Code Exploration}
\label{alg:code_exploration}
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}

\Input{Code graph $G$, query context $Q$, budget $B$}
\Output{Exploration sequence $(v_1, v_2, \ldots, v_k)$}

Initialize priority queue $\mathcal{Q}$ with seed nodes\;
Set visited = $\emptyset$, budget remaining $b = B$\;

\While{$\mathcal{Q} \neq \emptyset$ and $b > 0$}{
    $v = \mathcal{Q}.\text{pop()}$ (highest priority)\;
    
    \If{$v \notin \text{visited}$}{
        Visit node $v$, add to exploration sequence\;
        $\text{visited} = \text{visited} \cup \{v\}$\;
        $b = b - \text{cost}(v)$\;
        
        \For{each neighbor $u \in N(v)$}{
            $\text{priority}(u) = \text{Utility}(u | Q) + \text{Novelty}(u | \text{visited})$\;
            $\mathcal{Q}.\text{insert}(u, \text{priority}(u))$\;
        }
    }
}

\Return{exploration sequence}\;
\end{algorithm}

\section{Probabilistic Code Generation Model}

Code generation is modeled as constrained sequence generation with quality objectives.

\subsection{Generation Framework}

\begin{definition}[Constrained Generation Problem]
\label{def:constrained_generation}
Generate code sequence $c = (c_1, c_2, \ldots, c_n)$ to maximize:
\begin{equation}
\text{Score}(c) = \log P(c | \text{intent}, \text{context}) + \lambda \cdot \text{Quality}(c)
\end{equation}
subject to constraints $c \in \mathcal{C}$ where $\mathcal{C}$ represents syntactic, semantic, and style constraints.
\end{definition}

\begin{definition}[Quality Decomposition]
\label{def:quality_decomposition}
The quality function decomposes as:
\begin{align}
\text{Quality}(c) = &w_1 \cdot \text{Correctness}(c) + w_2 \cdot \text{Readability}(c) \\
&+ w_3 \cdot \text{Efficiency}(c) + w_4 \cdot \text{Maintainability}(c)
\end{align}
\end{definition}

\subsection{Constrained Decoding}

\begin{algorithm}
\caption{Constrained Beam Search for Code Generation}
\label{alg:constrained_beam_search}
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}

\Input{Intent $I$, context $C$, constraints $\mathcal{C}$, beam size $k$}
\Output{Generated code sequence $c^*$}

Initialize beam $B = \{[\text{START}]\}$\;

\For{position $t = 1$ to $\text{max\_length}$}{
    $\text{candidates} = \emptyset$\;
    
    \For{each sequence $s \in B$}{
        \For{each token $v \in \text{Vocabulary}$}{
            $s' = s + [v]$\;
            \If{$s'$ satisfies constraints in $\mathcal{C}$}{
                $\text{score}(s') = \log P(s' | I, C) + \lambda \cdot \text{Quality}(s')$\;
                $\text{candidates} = \text{candidates} \cup \{s'\}$\;
            }
        }
    }
    
    $B = \text{top-}k(\text{candidates})$ by score\;
    
    \If{all sequences in $B$ are complete}{
        break\;
    }
}

\Return{$c^* = \arg\max_{s \in B} \text{score}(s)$}\;
\end{algorithm}

\subsection{Quality Estimation}

\begin{definition}[Multi-dimensional Quality Assessment]
\label{def:quality_assessment}
Quality assessment uses multiple specialized models:
\begin{align}
\text{Correctness}(c) &= P(\text{passes tests} | c) \cdot P(\text{type checks} | c) \\
\text{Readability}(c) &= \text{StyleScore}(c) \cdot \text{ComplexityScore}(c) \\
\text{Efficiency}(c) &= \text{TimeComplexity}(c)^{-1} \cdot \text{SpaceComplexity}(c)^{-1}
\end{align}
\end{definition}

\section{Risk Assessment and Safety Models}

Safety considerations are modeled probabilistically with chance constraints.

\subsection{Risk Quantification}

\begin{definition}[Risk Categories]
\label{def:risk_categories}
We identify several categories of risk:
\begin{itemize}
    \item $R_{\text{break}}$: Probability of breaking existing functionality
    \item $R_{\text{security}}$: Probability of introducing security vulnerabilities  
    \item $R_{\text{data}}$: Probability of data loss or corruption
    \item $R_{\text{performance}}$: Probability of performance degradation
\end{itemize}
\end{definition}

\begin{definition}[Composite Risk Function]
\label{def:composite_risk}
The overall risk is computed as:
\begin{equation}
R(s, a) = 1 - \prod_{i} (1 - R_i(s, a))
\end{equation}
assuming independence of risk factors.
\end{definition}

\subsection{Risk-Aware Decision Making}

\begin{definition}[Chance Constrained Optimization]
\label{def:chance_constrained}
The risk-aware optimization problem is:
\begin{align}
\max_{a} \quad &\E[\text{Utility}(s, a)] \\
\text{subject to} \quad &P(R(s, a) > r^*) \leq \epsilon
\end{align}
for risk threshold $r^*$ and confidence level $\epsilon$.
\end{definition}

\begin{algorithm}
\caption{Risk-Aware Action Selection}
\label{alg:risk_aware_selection}
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}

\Input{State $s$, candidate actions $\mathcal{A}$, risk threshold $r^*$, confidence $\epsilon$}
\Output{Selected action $a^*$}

$\text{safe\_actions} = \emptyset$\;

\For{each action $a \in \mathcal{A}$}{
    Estimate risk: $\hat{R}(s, a) = \sum_i w_i R_i(s, a)$\;
    Compute confidence interval: $[\hat{R}(s, a) - \delta, \hat{R}(s, a) + \delta]$\;
    
    \If{$\hat{R}(s, a) + \delta \leq r^*$}{
        $\text{safe\_actions} = \text{safe\_actions} \cup \{a\}$\;
    }
}

\If{$\text{safe\_actions} = \emptyset$}{
    \Return{$a_{\text{safest}} = \arg\min_{a \in \mathcal{A}} \hat{R}(s, a)$}\;
}
\Else{
    \Return{$a^* = \arg\max_{a \in \text{safe\_actions}} \E[\text{Utility}(s, a)]$}\;
}
\end{algorithm}

\section{Online Learning and Adaptation}

The system continuously adapts its models based on observed outcomes.

\subsection{Multi-Armed Bandit with Side Information}

\begin{definition}[Tool Performance Learning]
\label{def:tool_learning}
Tool effectiveness is learned online using:
\begin{equation}
\hat{\mu}_{t,\theta}^{(n)} = \frac{1}{n} \sum_{i=1}^{n} r_i^{(t,\theta)} + \text{side\_info\_adjustment}
\end{equation}
where side information includes tool documentation, similar tool performance, and theoretical analysis.
\end{definition}

\subsection{Adaptive Parameter Tuning}

\begin{definition}[Online Gradient Descent]
\label{def:online_gradient}
System parameters are updated using online gradient descent:
\begin{equation}
\theta_{t+1} = \theta_t - \eta_t \nabla_{\theta} \ell_t(\theta_t)
\end{equation}
where $\ell_t$ is the instantaneous loss and $\eta_t$ is the learning rate.
\end{definition}

\begin{theorem}[Convergence of Adaptive Parameters]
\label{thm:adaptive_convergence}
Under standard assumptions on the loss function and learning rates, the adaptive parameters converge to a stationary point of the expected loss.
\end{theorem}

\section{Complexity Analysis}

\subsection{Time Complexity Results}

\begin{theorem}[Tool Selection Complexity]
\label{thm:tool_selection_complexity}
The LinUCB tool selection algorithm requires $O(d^3 + k \cdot d^2)$ operations per decision, where $d$ is the feature dimension and $k$ is the number of tools.
\end{theorem}

\begin{theorem}[Context Selection Complexity]
\label{thm:context_selection_complexity}
The greedy context selection algorithm runs in $O(n^2 \log n)$ time for $n$ candidate elements under the submodular assumption.
\end{theorem}

\begin{theorem}[Code Exploration Complexity]
\label{thm:exploration_complexity}
Priority-based code exploration requires $O(V \log V + E)$ operations where $V$ is the number of nodes and $E$ is the number of edges in the code graph.
\end{theorem}

\subsection{Space Complexity Results}

\begin{proposition}[Memory Requirements]
\label{prop:memory_requirements}
The system's memory usage scales as:
\begin{itemize}
    \item Tool models: $O(k \cdot d^2)$ for $k$ tools and $d$ features
    \item Context storage: $O(B)$ for budget $B$
    \item Code graph: $O(V + E)$ for graph storage
    \item Belief state: $O(d)$ for compressed representation
\end{itemize}
\end{proposition}

\section{Summary}

This chapter has presented detailed mathematical models for the core algorithmic components of \ClaudeCode{}. Key contributions include:

\begin{itemize}
    \item Contextual bandit framework for tool selection with regret bounds
    \item Submodular optimization approach for context management with approximation guarantees
    \item Queueing-theoretic analysis of concurrent tool execution
    \item Graph-based models for code understanding and exploration
    \item Probabilistic framework for constrained code generation
    \item Risk assessment models with chance constraints
    \item Online learning algorithms with convergence guarantees
    \item Comprehensive complexity analysis
\end{itemize}

These mathematical models provide both theoretical foundations and practical algorithmic guidance for implementing AI-powered code intelligence systems. The models balance mathematical rigor with computational tractability, enabling both theoretical analysis and practical deployment.

In the next chapter, we build upon these models to develop a comprehensive algorithmic analysis framework, providing deeper theoretical insights and performance guarantees.